Remark 4.2 Lemma REF  is a continuous-space version of
[1]} and its proof follows similar lines.
We will only need part (b) of this lemma in this paper,
for the proof of Theorem REF . Part (a) is required in
[2]}, where an invariance principle for \(\overline{m}(t)\) 
is proved. However, the proof of Lemma REF (b)
heavily builds on that of (a), which is why it is natural to provide it
here.

